Identifiability condition

In mathematics, the identifiability condition is defined as

:$f(x)\; =\; f(y)\; Leftrightarrow\; x\; =\; y\; quad\; forall\; x,y$

which says that if a function evaluates the same, then the arguments must be the same. I.e., a function is "identifiable" iff it is one-to-one. The article on injective functions deals with this same topic more abstractly.

Example 1

Let the function be the sine function.This function does not satisfy the condition when the parameter is allowed to take any real number.

:$sin(0)\; =\; 0\; =\; sin(2pi)\; =\; sin(4pi)\; =\; cdots$

However, if the parameter is restricted to $[-pi/2,\; pi/2]$ then it satisfies the condition.

Example 2

Let the function be $y\; =\; f(x)\; =\; x^3$.This function clearly satisfies the condition as it is a one-to-one function.

Example 3

Let the function be the normal distribution with zero mean.For a fixed random value and nonfixed variance, this function satisfies the condition

:$f(x;\; sigma\_1^2)\; =\; f(x;\; sigma\_2^2)\; Leftrightarrow\; sigma\_1^2\; =\; sigma\_2^2$